This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1828 Excerpt: ...a sum 2a, with the nth part of what remains, when the first share and 2a have been taken away; to the third, a sum 3a, with the nth part of the new remainder; and so on. Moreover, the shares of all the sons are equal. What is the property of the father, the share of each of the children, and the number of children? ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1828 Excerpt: ...a sum 2a, with the nth part of what remains, when the first share and 2a have been taken away; to the third, a sum 3a, with the nth part of the new remainder; and so on. Moreover, the shares of all the sons are equal. What is the property of the father, the share of each of the children, and the number of children? This problem is remarkable, since it contains more conditions than are necessary to determine the value of the unknown quantities. Let x be the property of the father; a---a is what remains when a is taken away. Therefore the share of the eldest is x--a na + x--a a H, or n n The subtraction of this part and of 2a from the whole, leaves-an+x--a nx--3an--x+a x--2a, or, n n, ., . nx--3an--x+a of which the nth part is. n2 Then the share of the second is n nx--3an--x+a 2an1+nx--3an--x+a 2a +, or In the same way the other shares may be found; but as they must all be equal, the equation of the problem may be formed by simply equating the two first shares, which gives an + a, '--a _ 2an2 + nx--3an--x+a n n2' whence x = atfl--2an + a. Substitute this value of x in the first share, which then becomes cn+an2--2an + a--a an2--an..., or, or an--a, or a(n--J); n n and as all the shares must be equal, the division of the whole property by the share of the first gives the number of sons; this. an1--2an + a., number is, or n--1. an--a The whole property is, therefore, a(n--l)2. The share of each son is... a(n--1). The number of sons is... n--1. It remains to be found whether the other conditions of the problem are satisfied; that is, whether, if the second have 2a with the nth part of the remainder, the third 3a, with the nth part of the remainder, the share of each is a(n--1). The difference between the whole property and the first part is a(n--I)2--a(n--1), and..
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